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The Republican presidential candidates returned to the campaign trail today after squaring off last night on a debate stage in South Carolina.

Front-runners Donald Trump and Ted Cruz dominated the event, while the rest of the field continue to try to keep pace.

NewsHour correspondent William Brangham reports.

SEN. TED CRUZ (R-TX), Republican Presidential Candidate: My friend Donald said that he had had his lawyers look at this from every which way, and there was no issue there.

WILLIAM BRANGHAM:

The issue is Ted Cruz's eligibility to run for the presidency. And Donald is Donald Trump, who's been highlighting the fact that Cruz was born to an American mother, but in Canada. Early on, the issue ignited a testy exchange between the two candidates leading the Republican field in Iowa.

SEN. TED CRUZ:

I recognize that Donald is dismayed that his poll numbers are falling in Iowa. But the facts and the law here are really quite clear. Under longstanding U.S. law, the child of a U.S. citizen born abroad is a natural-born citizen.

WILLIAM BRANGHAM:

Trump, in turn, insisted the issue would hang over his rival's head until it's put to rest.

DONALD TRUMP, Republican Presidential Candidate:

And if you become the nominee, who the hell knows if you can even serve in office? So you should go out, get a declaratory judgment, let the courts decide.

WILLIAM BRANGHAM:

That set the tone for a rough-and-tumble evening, and the candidates got into it again over the Texas senator's statements that Trump embodies — quote — "New York values."

SEN. TED CRUZ:

And listen, there are many, many wonderful, wonderful working men and women in the state of New York. But everyone understands that the values in New York City are socially liberal or pro-abortion or pro-gay-marriage, focus around money and the media.

WILLIAM BRANGHAM:

Trump answered that with a spirited defense of his hometown, harking back to the days after 9/11.

DONALD TRUMP:

And we rebuilt downtown Manhattan, and everybody in the world watched and everybody in the world loved New York and loved New Yorkers. And I have to tell you, that was a very insulting statement that Ted made.

(CHEERING AND APPLAUSE)

WILLIAM BRANGHAM:

But it wasn't just a two-man show. Cruz took aim at Florida Senator Marco Rubio for his involvement in a 2013 immigration reform compromise.

SEN. TED CRUZ:

It is also the case that that Rubio-Schumer amnesty bill, one of the things it did is it expanded Barack Obama's power to let in Syrian refugees. It enabled hi, the president, to certify them en masse without mandating meaningful background checks.

WILLIAM BRANGHAM:

Rubio counterattacked with a charge that Cruz himself has gone back and forth on a number of issues.

SEN. MARCO RUBIO (R-FL), Republican Presidential Candidate: Ted Cruz, you used to say you supported doubling the number of green cards. Now you say that you're against it. You used to support a 500 percent increase in the number of guest workers. Now you say that you're against it.

I saw you on the Senate floor flip your vote on crop insurance because they told you it would help you in Iowa, and last week, we all saw you flip your vote on ethanol in Iowa for the same reason.

(CHEERING AND APPLAUSE)

WILLIAM BRANGHAM:

Rubio also sought to take down New Jersey Governor Chris Christie, a rival for support of the party's so-called establishment wing, by painting his record as liberal.

SEN. MARCO RUBIO:

Governor Christie has endorsed many of the ideas that Barack Obama supports, whether it is Common Core or gun control or the appointment of Sonia Sotomayor or the donation he made to Planned Parenthood.

WILLIAM BRANGHAM:

Christie, in turn, dismissed the jibe as all politics, and no principle.

GOV. CHRIS CHRISTIE (R-NJ), Republican Presidential Candidate: Two years ago, he called me a conservative reformer that New Jersey needed. That was before he was running against me. Now that he is, he's changed his tune.

WILLIAM BRANGHAM:

But Rubio and Christie saved their toughest words for the president, hammering away all evening.

GOV. CHRIS CHRISTIE:

Mr. President, we're not against you. We're against your policies. And we are going to kick your rear end out of the White House come this fall.

(CHEERING AND APPLAUSE)

WILLIAM BRANGHAM:

Ohio Governor John Kasich and former Florida Governor Jeb Bush also kept most of their fire trained on the White House. And Bush emerged today with a high-profile endorsement from South Carolina Senator Lindsey Graham, who recently gave up his own presidential bid. The South Carolina primary is February 20.

First, though, the Republicans enter the home stretch in Iowa and New Hampshire. Their next debate will be in Iowa on January 28, the Thursday before the caucuses.